1. Mathematics I. Lectures BSc materials
Topic 4
Infinite sequences j
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2. The concept of an infinite sequence
Definition: An infinite sequence of real numbers is a function whose domain is the set of positive integers.
Definition: A sequence is a mapping which maps the elements of positive integers to the element of real numbers.
The real number an is called the nth term of the sequence which is assigned to the positive integer n.
Examples:
The sequence designation is {an} .
There are infinite terms of a sequence.
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3. The methods of defining sequences
(1) With a general term. Example each terms of the sequence can be calculated directly
(2) With instructions. Example
(3) With a recurrent relation. Example a1=1, a2=1 ,an=an-1+an n≥3
1,1,2,3,5,8,13…
Fibonacci-sequence
How many pairs of rabbits come from a single pair in a year if every month each pair gives birth to a new pair which from the second month will be capable of breeding and any offspring does not die?
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4. Representing sequences graphically
In the coordinate system of xy-plane
On a real line an a1 a4 a3 a2
A sequence is represeted with a set of discrete points. n
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5. Representing sequences graphically
In a coordinate system
On a real line
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Properties of sequences
Defintion. A sequence is bounded bellow if there is a number k that is smaller than any member of the sequence.
Definition. A sequence is bounded above if there is a number K that is greater than any member of the sequence.
Definition. A sequence is called bounded if it is bounded bellow and above.
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Properties of sequences
Definition. A sequence is strictly monotonically increasing if
an<an+1
Definition. A sequence is monotonically increasing if
Definition. A sequence is strictly monotonically decreasing, if
Definition. A sequence is monotonically decreasing, if
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Properties of sequences
Definition: Number x is approximately equal to number A with an error less then ε>0 if or equivalently
Definition: The neighbourhood of number A with a radius e>0 is called the open interval (A-e, A+e). a5 a4 a3 a2 a1 b1 b3 b5 b6 b4 b2
c1=c3=c5=…. c8 c6 c4 c2
d1=d3=d5=….
d2=d4=d6=…. e1 e2 e3 e4 e5 e6 e7
0-e
0+e
The terms belong to the neighbourhood 0 with radius 1/3 for the sequence {an} , if n>3, for the sequence {bn} if n>3, for the sequence {cn}, if n>6.
For the sequences {an}, {bn}, {cn}
only finite terms are left out from the neighbourhood of 0 with radius ε
almost all the terms are in the neighbourhood of 0 with radius ε
terms from some index are approximating 0 with error less than ε.
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Properties of sequences
For all positive e almost all terms of the sequences {an}, {bn} and {cn} are in the neighbourhood of 0 with radius e. If value e decreases, there is an increase in the number of terms of sequences {an}, {bn}, {cn} which are left out from the neighbourhood of 0 with radius e.
Definition 1. The sequence {an} is convergent and the limit equals to the number A, if for all positive ε > 0 there is a (no is the threshold index which may depend on ε ) such that when n>no .
With signals (limit an equals to A as n tends to infinity) or
Definition 2. The sequence {an} is convergent and the limit equals to A, if finite terms of the sequence are left out from the environment of number A for all radius ε>0.
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10. Convergence of the sequence 1/n
Let us prove by using the definition of convergence of the infinite sequences that
How many terms of the sequence are left out from the which are left out from the neighbourhood of the limit with radius e=0,03, i.e. find no(e)=?
The transformations are equivalent  the reverse transformations are also fulfilled.
So if n > no (0,03)= 33, then
i.e. the sequence is convergent and the limit is 0.
33 terms of the sequence are left out from the neighbourhood 0 with radius ε=0.03.
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11. Elementary rules for calculating limits
Theorem If a sequence converges, then its limit is unique.
Theorem If an → a, bn → b, then 1) 2) 3) 4) if 5)
Theorem ( Sandwich rule) If an → a, bn → a,
, then cn → a
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Divergent sequences
Definition: Non-convergent sequences are called divergent.
Definition: The sequence {an} tends to +∞, if for all „big” number K there is a no(K) (threshold index which may depend on K), such that
With symbols:
Definition: , if
, if n > no
Remark. If a sequence tends to +∞ or -∞, then the sequence is called convergent in a broader sense.
Summary. A sequence called convergent when the limit exists and finite.
A sequence is called divergent when the limit +∞, or -∞ or does not exist. .
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Divergent sequences ,
Theorem If an > 0 and an → 0, then
Theorem If
, then
(i)
(ii)
(iii)
(iv)
Theorem If
,then
(i)
, (ii)
Remark If
then there is nothing to prove about
and
Remark If
, then there is nothing to prove about .
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14. Relations between boundedness, monotonicity and convergence
Theorem A convergent sequence is bounded.
(Convergence  boundedness. Boundedness is a necessary condition for convergence)
Proof Let
From the definiton of convergence for arbitrary e>0
so that, if n > no 
that is
Let
Therefore
bounded
Remark The converse of the theorem is not true e.g.
Theorem All monotone and bounded sequences is convergent. S
Monotonicity and boundedness  convergence
Monotonicity and boundedness are sufficient conditions of convergence M B C
S={sequences}
B={bounded}
M={monotonic}
C={convergent}
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15. Commonly occurring limits, 1) 2)
geometric sequence 3)
a > 0 .
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16. Commonly occurring limits, 4) 5)
a > 0 6) .
e is the Euler-number (irrational number)
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